Kink dynamics in spatially inhomogeneous media: the role of internal modes

Abstract

The phenomenon of length-scale competition in soliton-bearing equations perturbed by spatially dependent terms A. Sánchez and A. R. Bishop, SIAM Rev. 40, 579 1998 is analyzed from a general viewpoint. We show that the perturbation gives rise to an effective potential for solitons, which consists of wells and barriers. We calculate the effect of these potential features on the solitons, establishing a direct relationship between the maxima, minima, and curvature of the potential with soliton deformations. When the typical wavelength of the perturbation is of the order of the soliton width, these deformations are seen to correspond to the excitation of shape modes and can lead to the dissipation of the soliton kinetic energy and, further, to the impossibility of soliton propagation. Thus, we demonstrate that the mechanism for length-scale competition is related to changes in the dynamics of the internal modes. We study different examples where the perturbation is introduced parametrically and nonparametrically to make it clear that our results apply to a wide class of equations.